CUET Mathematics: CalculusContinuity & Differentiability. Free, no login required.

If $f(x) = \begin{cases} \frac{k\cos x}{\pi - 2x}, & x \neq \frac{\pi}{2} \\ 3, & x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$, then k is :

The derivative of $\sec(\tan \sqrt{x})$ with respect to x is :

If $y = \log\left[\frac{x^2}{e^2}\right]$ then value of $\frac{d^2y}{dx^2}$ is :

The condition on a and b, such that for $y = \frac{a}{x} - \frac{b}{x^2}$, $\frac{dy}{dx} = 0$ at $x=1$ is :

The points of discontinuity of the function f defined by $f(x) = \begin{cases} x+2 & x \leq 1 \\ x-2 & 1 < x < 2 \\ 0 & x \geq 2 \end{cases}$ are :

If $\cos y = x\cos(a+y)$, then $\frac{dy}{dx} =$

If $f(x) = \begin{cases} \frac{x^2 - 9}{x - 3}, & x \neq 3 \\ 5, & x = 3 \end{cases}$ then $f(x)$ :

If $y = x^x$, then value of $\frac{dy}{dx}$ at $x = 2$ is :

A function $f(x)$ is defined by :

$f(x) = \begin{cases} x + 2, & \text{if } x < 0 \\ -x + 2, & \text{if } x > 0 \end{cases}$

Which of the following is true ?

If $y = 500 e^{7x} + 600 e^{-7x}$ and $\frac{d^2 y}{dx^2} = ky$, then the value of k is :

If $y = x^{\sin x}$, then value of $x^{-\sin x} \frac{dy}{dx} - \cos x \log x$ is :

$f(x) = [x]$, where $[\,]$ represents greatest integer function.

(A) For $2 \leq x < 3$, $[x] = 3$

(B) For $2 \leq x < 3$, $[x] = 2$

(C) Right hand derivative of f at $x = 2$ is not defined

(D) Left hand derivative of f at $x = 2$ is zero

(E) $f(x)$ is not differentiable at $x = 2$

Choose the correct answer from the options given below :

If $x^y = e^{x-y}$, then $\frac{dy}{dx} =$

Let $y = \log(x + \sqrt{x^2+1})$, and $a\frac{d^2y}{dx^2} + b\frac{dy}{dx} = c$. Then identify the correct statements about the values of $a$, $b$ and $c$ :

(A) $a = 1 + x^2$

(B) $b = 0$

(C) $c = 0$

(D) $b = x$

(E) $c = 2$

Choose the correct answer from the options given below :

If $\sqrt{y+x} + \sqrt{y-x} = a$, $a > 1$, $\frac{d^2y}{dx^2}$ is equal to :

If $x = \int_0^y \frac{dt}{\sqrt{1+9t^2}}$ and $\frac{d^2y}{dx^2} = \lambda y$, then, $\lambda$ is equal to

The value of k, for which the function $f(x) = \begin{cases} \frac{\sin kx}{x} + 3\cos x, & x \neq 0 \\ 7, & x = 0 \end{cases}$ is continuous at $x = 0$, is

If $f$ is a function defined by $f(x) = \begin{cases} 5x^2 - x + 3, & x < 1 \\ 3x + 4, & x \geq 1 \end{cases}$, then, at $x = 1$, $f$ is

If $y = \frac{\log_e x}{x}$, then $\frac{d^2y}{dx^2} =$

If $y = \frac{\sqrt{x+1} + \sqrt{x-1}}{\sqrt{x+1} - \sqrt{x-1}}$, then $(x^2-1)^{3/2} \frac{d^2y}{dx^2} =$

If $f(x) = \begin{cases} \frac{\tan(\pi/4 - x)}{\cot 2x}, & x \neq \pi/4 \\ k, & x = \pi/4 \end{cases}$ is continuous at $x = \pi/4$, then the value of k is

The derivative of $\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ w.r.t. $\tan^{-1}\left(\frac{2x}{1-x^2}\right)$ is

Derivative of $x^3 + 1$ with respect to $x^2 + 1$ is

If $x = 2\sin\theta$ and $y = 2\cos\theta$, then the value of $\frac{d^2y}{dx^2}$ at $\theta = 0$ is

If $x = e^{y + e^{y + e^{y + \ldots \infty}}}$, $x > 0$, then $\frac{dy}{dx}$ is equal to

The function $f(x) = e^{|x|}$ is

(a) continuous everywhere on $R$

(b) not continuous at $x = 0$

(c) Differentiable everywhere on $R$

(d) not differentiable at $x = 0$

(e) continuous and differentiable on $R$

Choose the most appropriate answer from the options given below :

Let $y = m\sin rx + n\cos rx$. What is the value of $\frac{d^2y}{dx^2}$?

Match List I with List II

Choose the correct answer from the options given below:

The function $f(x) = |x - 1|$ is